Detectors are relatively cheap and come out of the box for citizen science projects, this project is relatively well known: http://www.cosmicwatch.lns.mit.edu/
For high energy cosmic rays, you need to observe many of these secondary particles that are produced during the absorption with high temporal resolution, few nanoseconds, to be able to tell anything about their properties.
There are essentially five or so detection principles for measuring cosmic rays at the ground, and many observatories combine multiple techniques.
First, you can observe the secondary particles that reach the ground, mainly muons and electrons. This is possible either using water tanks with photosensors inside detecting Cherenkov light or using scintillators like in the project I linked above.
Then you can also detect very short and faint light that is also emitted in the air shower. This also comes in two variants: Cherenkov light is emitted in a cone around the charged particles and results in a "light pool" of roughly 250m diameter on the ground. Fluorescence light is emitted in all directions. We build optical telescopes with extremely fast and sensitive cameras to detect Cherenkov or fluorescence light.
Last, there is also radio emission from air showers, you can detect with antennas.
Auger in Argentina combines water tanks, scintillators and fluorescence telescopes and is investigating adding in radio antennas.
Telescope Array, the experiment which measured this event here, is using scintillators and fluorescence telescopes.